Can someone solve this, i have tried for 2 days!!

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obelisk1151

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Unfortunately its on german and my english ins't that good so i've translated it over google translate:
Given are the 2 × 3 matrices A1 = (1 0 1 0 1 2), A2 = (1 0 1 3 1 0),A3 = (5 0 5 6 5 6).
(a) Are these matrices of R2 × 3 linearly dependent? If yes, represent A3 as a linear combination of the other two.
(b) Does the span of the matrices {A1, A2, A3} form a subspace of R2 × 3? If so, what is the dimension of this subspace?
(c) What is the dimension of the R-vector space R2 × 3? Enter a base.
(d) Which dimension has the R-vector space C2 × 3? Enter a base.


I have already done the (a) part, but i don't know how do to the rest, pls someone help me.
 
obelisk1151 said:
View attachment 14732
Unfortunately its on german and my english ins't that good so i've translated it over google translate:
Given are the 2 × 3 matrices A1 = (1 0 1 0 1 2), A2 = (1 0 1 3 1 0),A3 = (5 0 5 6 5 6).
(a) Are these matrices of R2 × 3 linearly dependent? If yes, represent A3 as a linear combination of the other two.
(b) Does the span of the matrices {A1, A2, A3} form a subspace of R2 × 3? If so, what is the dimension of this subspace?
(c) What is the dimension of the R-vector space R2 × 3? Enter a base.
(d) Which dimension has the R-vector space C2 × 3? Enter a base.


I have already done the (a) part, but i don't know how do to the rest, pls someone help me.
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Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please share your work/thoughts about this assignment.

Please show us the work you have done to solve part (a).
 
Well you got me curious. Are the three matrices independent or not? Don't you think that answer would help for some/all the other parts.
 
Jomo said:
Well you got me curious. Are the three matrices independent or not? Don't you think that answer would help for some/all the other parts.
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I wanted to sent the work ive done (a), but after checking it i have realized that i've screwed up.
 
The first question asks whether \(\displaystyle \begin{pmatrix}5 & 0 & 5 \\ 6 & 5 & 6\end{pmatrix}\) is a linear combination of \(\displaystyle \begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}\) and \(\displaystyle \begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}\).

That is, do there exist a and b such that \(\displaystyle a\begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}+ b\begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}= \begin{pmatrix}a+b & 0 & a+b \\ 3a & a+b & 2b\end{pmatrix}= \begin{pmatrix}5 & 0 & 5 \\ 6 & 5 & 6\end{pmatrix}\).

And that is the same as asking whether there exist a and b such that a+ b= 5, 3a= 6, 2b= 6. If there are what are a and b?

But it is easy to see that the first two matrices are linearly independent since one is not a multiple of the other. So the subspace spanned by those two matrices is 2 dimensional, all such matrices being of the form \(\displaystyle a\begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}+ b\begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}\).

"\(\displaystyle R_{2x3}\)", the space of all 2 by 3 matrices, consists of all matrices of the form \(\displaystyle \begin{pmatrix} a & b & c \\ d & e & f\end{pmatrix}\). That contains 6 independent numbers so is of dimension 6. Find each of the 6 basis matrices by taking each of a, b, c, d, e, and f to be 1 and the others 0. Two such basis matrices are \(\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\) and \(\displaystyle \begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}\)

For the last one, every complex number can be constructed from two real numbers so \(\displaystyle C_{23}\) has dimension 2(6)= 12. There are 12 basis matrices. Two of them are Two such basis matrices are \(\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\) and \(\displaystyle \begin{pmatrix}i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\).
 
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HallsofIvy said:
The first question asks whether \(\displaystyle \begin{pmatrix}5 & 0 & 5 \\ 6 & 5 & 6\end{pmatrix}\) is a linear combination of \(\displaystyle \begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}\) and \(\displaystyle \begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}\).

That is, do there exist a and b such that \(\displaystyle a\begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}+ b\begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}= \begin{pmatrix}a+b & 0 & a+b \\ 3a & a+b & 2b\end{pmatrix}= \begin{pmatrix}5 & 0 & 5 \\ 6 & 5 & 6\end{pmatrix}\).

And that is the same as asking whether there exist a and b such that a+ b= 5, 3a= 6, 2b= 6. If there are what are a and b?

But it is easy to see that the first two matrices are linearly independent since one is not a multiple of the other. So the subspace spanned by those two matrices is 2 dimensional, all such matrices being of the form \(\displaystyle a\begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 2\end{pmatrix}+ b\begin{pmatrix}1 & 0 & 1\\ 3 & 1 & 0\end{pmatrix}\).

"\(\displaystyle R_{2x3}\)", the space of all 2 by 3 matrices, consists of all matrices of the form \(\displaystyle \begin{pmatrix} a & b & c \\ d & e & f\end{pmatrix}\). That contains 6 independent numbers so is of dimension 6. Find each of the 6 basis matrices by taking each of a, b, c, d, e, and f to be 1 and the others 0. Two such basis matrices are \(\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\) and \(\displaystyle \begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}\)

For the last one, every complex number can be constructed from two real numbers so \(\displaystyle C_{23}\) has dimension 2(6)= 12. There are 12 basis matrices. Two of them are Two such basis matrices are \(\displaystyle \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\) and \(\displaystyle \begin{pmatrix}i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}\).
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a) didn't the first question ask me if all 3 matrices are linearly dependent?
and if they are then to represent a3 as a linear combination of a2 and a1
b) its asking if the span of all 3 matrices form a subspace of R2+3 and what dimension that subspace has.
 
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obelisk1151 said:
a) didn't the first question ask me if all 3 matrices are linearly dependent?
and if they are then to represent a3 as a linear combination of a2 and a1
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Yes, you are correct and HallsofIvy did just that. Halls showed that aA1 + bA2 = A3 under certain constraints on a and b. Were you able to solve for a and b?
 
Jomo said:
Yes, you are correct and HallsofIvy did just that. Halls showed that aA1 + bA2 = A3 under certain constraints on a and b. Were you able to solve for a and b?
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wait im confused i know that to prof that that they are linearly dependent its xA1+yA2+zA3=0 and if x,y,z are all 0 then the matrices are linearly independent otherwise if x,y,z are not all 0 then the matrices are linearly dependent
or im i wrong here?
aA1 + bA2 = A3 is to represent A3 as a linear combination of a2 and a1
 
obelisk1151 said:
wait im confused i know that to prof that that they are linearly dependent its xA1+yA2+zA3=0 and if x,y,z are all 0 then the matrices are linearly independent otherwise if x,y,z are not all 0 then the matrices are linearly dependent
or im i wrong here?
aA1 + bA2 = A3 is to represent A3 as a linear combination of a2 and a1
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You are correct that to show that they are linearly dependent you can conclude that xA1+yA2+zA3=0 and x, y and z are NOT all 0.
But is that the only way? If xA1+yA2+zA3=0 doesn't that imply (if z\(\displaystyle \neq\)0) A3 = (-x/z)A1 + (-y/z)A2????
 
Jomo said:
You are correct that to show that they are linearly dependent you can conclude that xA1+yA2+zA3=0 and x, y and z are NOT all 0.
But is that the only way? If xA1+yA2+zA3=0 doesn't that imply (if z\(\displaystyle \neq\)0) A3 = (-x/z)A1 + (-y/z)A2????
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alright but what about b)? can you explain that?
 
obelisk1151 said:
alright but what about b)? can you explain that?
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Does the span of the matrices {A1, A2, A3} form a subspace of R2 × 3? If so, what is the dimension of this subspace?
First you need to know what it means for a set of matrices to span all 2X3 matrices. It means any 2x3 matrix, A, can be expressed as a linear combination of A1, A2 and A3. Try this and show us your work.

Oh, and did you do the work for part a? Please show that first?
 
Jomo said:
Does the span of the matrices {A1, A2, A3} form a subspace of R2 × 3? If so, what is the dimension of this subspace?
First you need to know what it means for a set of matrices to span all 2X3 matrices. It means any 2x3 matrix, A, can be expressed as a linear combination of A1, A2 and A3. Try this and show us your work.

Oh, and did you do the work for part a? Please show that first?
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i did part a) and i got that -5z+5z=0,and it confused me
i did it with xA1+yA2+zA3=0
 
Jomo said:
Well why did you stop there?
-5z + 5z =0, So you have 0 = 0. What does that tell you? It should tell you that Z can be any number. Now finish up by figuring out what x and y equal and they my be in terms of z!
Continue!
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i dont know how to do it if z can be any number then how can i figure out y an x
 
obelisk1151 said:
i dont know how to do it if z can be any number then how can i figure out y an x
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Let z = k for some real number k
Then in terms of k what does x and y equal?

For example if r=3t, s=-2t and t=k, then the solution will be, r=3k, s=-2k and t=k. This means that there are an infinite number of solutions. So give us your final answer. That is tell us if the 3 matrices are independent or not and if they are dependent give us the linear combination that shows that.
 
but because there are an infinite number of solutions that means then that the matrices are dependent right?IMG_20191119_010210.jpg
 
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